Biomedical Engineering Reference

In-Depth Information

Fig. 10.4
Anti-bunching condition of a fibrillar structure. (
a
) Configuration of self-bunching in an

array of fibers distributed in (
b
)
square
,(
c
)
triangular
,or(
d
)
hexagonal
patterns

cause them to bundle together, as shown in Fig.
10.4a
. The stability condition can

be derived from the point of view of a maximum fiber length for spontaneous

separation of two fibers sticking together (e.g., [
7
]). In other words, given fiber

separation
w
and radius
R
, there exists a critical length
L
cr
beyond which lateral

bunching of neighboring fibers becomes stable configurations. Glassmaker et al.

[
33
] have derived the critical length for bunching of cylindrical fibers as

"

#
1
=
12

"

#
1
=
4

4
E
f
R

12
E
f
R
3
w
2

g
f

p

L
cr
¼

(10.5)

2
11

g
f
ð
1
n

2

f
Þ

Assuming that the fibers are distributed in a regular lattice pattern, one can relate

the fiber separation
w
, radius
R
to the area fraction

'

of a fiber array by

R ð
0

p

'
max
='

w ¼

1

<'<'
max
Þ

(10.6)

where

'
max
stands for the maximum area fraction of the given hair pattern. It can be

shown that

2
p
for a triangular lattice (Fig.
10.4b
),

'
max
¼ p=

'
max
¼ p=

4 for a

3
p
for a hexagonal lattice (Fig.
10.4d
). Inserting

square lattice (Fig.
10.4c
), and
p=

(
10.6
) into (
10.5
) leads to

1
=
3

1
=
2

p

'
max
='

E
f
R

g
f

L
cr
¼ Ra

1

(10.7)

1
=
12

3
3

4

p

where
a ¼

:

2
5

ð
1
n

2

f
Þ

Equation (
10.7
) has been derived for the lateral sticking between two neighboring

fibrils. Similar analysis can also be carried out for other possible bunching

configurations involving multiple neighboring fibers. We find that the critical fiber

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